© 2005 by CRC Press 53 3 Wave Propagation in Plane-Layered Media 3.1 REFLECTION AND REFRACTION OF PLANE WAVES AT THE BORDER OF TWO MEDIA Natural media — the atmosphere, earth, and others — can be supposed to be homogeneous only within bounded area of space. In reality, the permittivity of these media is a function of the coordinates and, in general terms, time, which is ignored, as a rule, because of the comparative sluggishness of natural processes. As usual, the time required for electromagnetic wave propagation in a natural medium is much less than typical periods of medium property changes. In the first approximation, natural media can be considered to be plane layered; that is, their permittivity changes in only one direction. If a Cartesian coordinates system is chosen in such a way that one of the coordinates (for example, z) coincides with this direction, then we may say that permittivity depends on this coordinate only. For the atmosphere of Earth, a concept of spherical-layered media would be more correct, and this idea is considered in the following chapters; however, we should point out that the curvature radius of media layers of the Earth is so large that the concept of plane stratification is sufficient in many cases. In this chapter, we will study only plane wave propagation in plane-layered media. We will assume waves as the plane only conventional because surfaces with constant phase and constant amplitude are not planes in the cases of wave propaga- tion direction inclined to the layers. On the whole, wave parameters cannot depend on only one particular Cartesian coordinate in all cases. In most of the cases that we will consider here, media parameters change little compared to wavelength; therefore, surfaces of equal phase or amplitude have sufficiently large curvature radii to be considered locally plane. It is quite acceptable to talk about plane waves in these terms. In some situations, medium properties can be changed sharply and on a wave- length scale; however, regions of such great change are usually rather thin layers and the waves are plane, outside of the layers just discussed. These layers correspond, in the extreme, to the areas jumpy changes of permittivity and occur at the boundary of two media. The air–ground boundary is an example of this idea. We turn now to consideration of plane wave reflection and transmission pro- cesses at a plane interface. We shall assume for the sake of simplicity that the permittivity of the medium from where the wave originates is equal to unity. Air is an example of such a medium. The problem of a wave incident on the ground from the air is investigated in such a way. The permittivity in this case is indicated by ε . The problem of wave reflection and transmission in media separated by a plane TF1710_book.fm Page 53 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 54 Radio Propagation and Remote Sensing of the Environment interface is well known and has been analyzed in many texts regarding electromag- netic waves; therefore, we will not investigate this problem in detail here and will examine only essential formulae. The electrical and magnetic fields of the incident wave are described by the vectors: (3.1) The boundary of the two media will be assumed to be a plane that is perpendicular to the z-axis. Let θ i represent the incidence angle (see Figure 3.1) in such a way that: (3.2) Note that vector q i can be defined as: (3.3) from which it follows that w i = k sin θ i . Let us call the field excited by the incident wave in the air the reflected wave and represent its fields by E r and H r . The wave in the ground is the refracted wave or transmitted wave , and the fields of this wave are represented by E t and H t . It is a simple matter to establish that the reflected and refracted waves are also found in the plane and that their wave vectors lie in the same plane as the vector of the incident wave. The wave vector of the reflected wave is: (3.4) and the wave vector of the refracted wave is: (3.5) The angle between vector q r and the z-axis is the angle of reflection . It is determined by the relation: (3.6) EE HH H qE qr qr ii i ii i iii ee k ii == =×     ⋅⋅ 0000 1 ,, ,EEqH iiii k k 00 1 = − ×     =,.q FIGURE 3.1 Plane wave incidence at the plane boundary. Z θ i q i θ j q j q t θ t cos .θ ii k = −⋅ () 1 eq z qw ewe ii ziz w= −− ⋅=k i 22 0,, qw e r 2 z w=+ − i k 2 , qw e t 2 z w= −− i kε 2 . cos ,θ rzr = ⋅ () 1 k eq TF1710_book.fm Page 54 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Wave Propagation in Plane-Layered Media 55 from which it follows that θ r = θ i ; that is, the angle of reflection equals the angle of incidence. In the case of complex permittivity ε , vector q r is complex and the transmitted wave is, in general, the inhomogeneous plane wave. The refraction angle determined from the equation: (3.7) is also complex in the general case. It is simple, however, to derive Snell’s law through the formula: (3.8) In the case of a weak absorptive medium, when we can neglect the imaginary part of ε , angle θ i is real, and we can fix the propagation direction of the refracted wave. Let us point out for later that: (3.9) The problem discussed here can be divided into the two cases of horizontal and vertical polarization. In the case of horizontal polarization, the amplitudes of the reflected and refracted waves are connected linearly with the amplitude of the incident wave: (3.10) where (3.11) is the coefficient of reflection for the horizontally polarized waves, and (3.12) is the coefficient of transmission. cosθ ε ε ti k k= −⋅ () = − 1 22 eq zt w εθ θsin sin . t = i qq eeq qq eq eq rzzt z z = −⋅ () = −⋅ () + − () + ⋅ iiii k21 2 , ε ii ()       2 e z . EEEE rh 0 th 000 ==FT ii ,, F ii ii h == −− + − cos sin cos sin θε θ θε θ 2 2 TF i ii hh =+ = + − 1 2 2 cos cos sin θ θε θ TF1710_book.fm Page 55 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 56 Radio Propagation and Remote Sensing of the Environment The magnetic and electrical wave components exchange places, in some sense, for the case of vertical polarization. In this case, the equations (3.13) are valid, where the reflection coefficient of the vertically polarized waves is: (3.14) and, correspondingly, the coefficient of transmission: (3.15) In these equations, the coefficients F h and F v are referred to as the Fresnel coefficients of reflection . They are complex values in the general case: , (3.16) and, therefore, wave reflection and refraction are accompanied not only by their amplitude change but also by phase rotation. Let us now express the fields of the reflected wave for the case of incidence on the interface the plane wave of any linear polarization. For convenience, we will set the vector of direction of the incident, reflected, and transmitted waves using the formulae: (3.17) and we can obtain the expressions: (3.18) (3.19) HH HH rv tv 00 00 ==FT ii ,, F ii ii v = −− + − εθ ε θ εθ ε θ cos sin cos sin , 2 2 TF i ii vv =+ = + − 1 2 2 εθ εθ ε θ cos cos sin . FFe FFe iF iF hh vv h v == arg arg , qeqeqe ii r kk k== =,,, rtt Eee eE ee ri i i F 0 2 0 2 112−⋅ ()       = ⋅ () −⋅ ()   zvz z     + ⋅ () {} + +× ⋅ eeee eeeH zz h ii zizi F (), Hee eH ee r 0 zhz z 112 2 0 2 −⋅ ()       = ⋅ () −⋅ ()   ii i F     + ⋅ () {} − − × ⋅ eeee eeeE zzii vz i z i F () TF1710_book.fm Page 56 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Wave Propagation in Plane-Layered Media 57 for the reflected wave. A simple expression for the field amplitudes: (3.20) can be established from the formulae provided above. Equations (3.18) and (3.19) can be reduced to: (3.21) (3.22) We can obtain similar formulae for the transmitted wave field. As was mentioned above, the processes of reflection and refraction are due to changes in the radiowave amplitude and phase. In particular, the tendency is for the reflected and refracted waves to become elliptically polarized by incidence on the interface the plane wave of arbitrary linear polarization. In other words, a change of the wave polarization takes place. As we already know, the polarization character can be described by a Stokes matrix. The processes of reflection and refraction can be considered as linear transforms; however, calculation of a Mueller matrix is rather a complicated procedure in this case. It is easier to do direct calculation of Stokes parameters. Let us represent the incident wave in the form , where the unitary vectors are directed toward the vectors of horizontal and vertical polarization and form the coordinate basis for the coordinate system relevant to the incident wave. We may use a the similar coordinate basis for the reflected wave and present its field in the form Now we will establish the relation between the orthogonal amplitude compo- nents of the reflected and incident waves. It is easy to do this for the horizontal components, where the required relation has the form As for the vertical components, it is necessary to note that in this case Let us represent the Stokes parameters of the incident wave by and the Stokes parameters of the reflected wave by Simple calculations allow us to set the relations between two systems of parameters and to establish the Stokes matrix transformation law for reflection of the wave. These relations have the form: EH 1 r 0 r 0 vz hz z () = () = ⋅ () + ⋅ () −⋅ 22 20 2 20 FF ii eE eH e ee i () 2 EE eE ee eee rh z zi zv zi 00 0 2 2 1 12=+ ⋅ () −⋅ () −⋅ () FF i i       −          ++ () ⋅ () } FFF hhvzii eee, HH H rv z zi zh zi 00 0 2 2 1 12=+ ⋅ () −⋅ () −⋅ () FF i i e ee eee       −          ++ () ⋅ () } FFF vhvzii eee. Ee e ihh vv =+EE ii ii() () () () ee hv and () ()ii Ee e rh rr v r v r =+EE h () () () () . EFE i h r hh () () .= EFE i v r vv () () .= − Sn n i() =÷ () 03 S n () . r TF1710_book.fm Page 57 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 58 Radio Propagation and Remote Sensing of the Environment (3.23a) (3.23b) (3.23c) (3.23d) Here, Θ r = arg F h – arg F v . Similar relations can be obtained for the refracted wave. Partial polarization appears at reflection of the noise radiation from the interface. In particular, the coefficient of polarization has the form: (3.24) Let us point out, in conclusion, that if it is a question of a wave incident on a medium with permittivity ε 1 at the border of a medium whose permittivity equals ε 2 , then in all previous formulae ε is equal to the relative permittivity ( ε = ε 2 / ε 1 ), and for wave number k it is necessary to substitute Let us also mention the specific relation connecting the reflection coefficients of vertically and horizontally polarized waves. The following relations are obtained from Equation (3.11): . By inserting these expressions into the formula for the reflection coefficient of the vertically polarized waves, we obtain the unknown relation: (3.25) SSFFSFF ii 00 22 1 22 1 2 r hv hv () () () =+ () + − ()       ,, SSFFSFF ii 10 22 1 22 1 2 r hv hv () () () = − () ++ ()       ,, SFFS S ii r232 r hv r () () () = − () sin cos ,ΘΘ SFFS S ii 332 r hv r r () () () = − + () cos sin .ΘΘ m FF FF = − + hv hv 22 22 . ε 1 k. εθ θε θ − = − + = + − + sin cos , cos 2 2 1 1 122 1 i h i i F F FF F hhh hh () 2 F FF F i i v hh h = − () − cos cos . 2 12 θ θ TF1710_book.fm Page 58 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Wave Propagation in Plane-Layered Media 59 3.2 RADIOWAVE PROPAGATION IN PLANE-LAYERED MEDIA Now, we will consider radiowave propagation in a medium whose permittivity is, in the general case, an arbitrary complex function of Cartesian coordinates; in this case, we choose z. As has been pointed out, such media are referred to as plane layered (stratified). It was noted, too, that Maxwell equations written in the form of Equations (1.91) to (1.92) are convenient to use. These equations are simplified essentially because ε = ε (z). Hence, it should be taken into account that a plane wave of any polarization propagates in one plane in this case and can be represented as the sum of two wave types. Let the y0z plane be the wave propagation plane, which means that the field does not depend on the x-coordinate and the operator ∂ / ∂ x = 0. Then, the basic waves are E-waves (or waves of horizontal polarization), for which the electrical field vector is directed perpendicularly to the plane of propagation (i.e., it is represented in the form ), and H-waves (or vertical polarized waves), for which . For E-waves, the field is described by the equation: (3.26) The equation for H-waves is more complicated: . (3.27) Equation (3.27) differs from the wave equation but is easily reduced to it by the substitution of : , (3.28) where the effective permittivity is determined by the equality: . (3.29) The solutions of Equations (3.26) and (3.28) we are seeking as: (3.30) Ee= Π ex He= Π mx ∂ ∂ ∂ ∂ ε 22 2 0 ΠΠ Π yz z 22 ++ () =k . ∂ ∂ ∂ ∂ ε ε∂ ∂ ε 22 2 1 0 ΠΠ Π Π m 2 m 2 m m xz zz z+ − + () = d d k ΠΠ mm = ε ˆ ∂ ∂ ∂ ∂ ε 22 2 0 ˆˆ ΠΠ Π m 2 m 2 m xz ++ () =kz e εε ε ε e k d d = −       2 2 1 z 2 Π y,z z y () = () Qe ik η . TF1710_book.fm Page 59 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 60 Radio Propagation and Remote Sensing of the Environment Then, the problem is reduced to solution of the common differential equation: (3.31) The constant of separation ( η ) is determined as follows. Let us suppose that ε changes with the z-coordinate beginning from some distance z 0 , and, at z z 0 , ε (z) = ε 0 = const . Let a plane wave described by exp[ ik (y sin θ i + zcos θ i )] be incident on the described medium from the area z < z 0 . Thus, (3.32) because the incident and exited waves are both matched dependent on y. Equation (3.31), in the general case, has no solution in the analytical form. It is expressed through known functions only in some cases with a particular view of the function ε (z). We will find one such partial solution in the next section. 3.3 WAVE REFLECTION FROM A HOMOGENEOUS LAYER To solve the problem of waves in plane-lay- ered media, let us first study the case of two media with permittivities ε 1 and ε 3 , separated by a homogeneous layer of thickness d , and with permittivity ε 2 (Figure 3.2). A layer of ice floating on water is an example of such a natural object. It is necessary to analyze two individual problems for E- and H-waves. Let us begin with the E-wave by assuming that the plane wave occurs in the semispace and the reflected wave appears as a result of inter- action with this layer. Let us represent poten- tial Π e in the form: (3.33) where θ i is the incident angle. We can now describe the function Q e (z) in the area as the sum: dQ d kQ 2 22 0 z z 2 + () −     = εη . ε 0 ηε θ = 0 sin i FIGURE 3.2 Plane wave propaga- tion in a homogeneous layer. d Z 1 ε 3 ε 2 ε −∞ <<z0 Π ee ik Qe i y,z z y () = () εθ 1 sin , −∞ <<z0 Qe Fe e ik e ik ii z zz () =+ − εθ εθ 11 cos cos . TF1710_book.fm Page 60 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Wave Propagation in Plane-Layered Media 61 The first term corresponds to the incident wave, for which the amplitude is assumed to be equal to unity. The second item corresponds to the reflected wave, and F e is the coefficient of reflection. The function Q e (z) satisfies the equation: inside the layer 0 < z < d and has the general solution: Finally, in the third medium (z > d ): , where T e is the coefficient of transmission. The values F e , T e , α , and β can be defined as solutions of equations derived from the boundary conditions. It is useful next to employ Snell’s law by introducing the angles θ 2 and θ 3 : (3.34) If ε 1 , ε 2 , and ε 3 are real numbers and if ε 2 – ε 1 sin 2 θ i > 0 and ε 3 – ε 1 sin 2 θ i > 0 (i.e., no total inner reflection), then these angles characterize the directions of the wave propagation in the media considered here. The boundary conditions, Equation (1.7), lead to continuity of function Π e and its first derivative over z at Four algebraic equations are obtained as a result: , Here, (3.35) dQ d kQ e ie 2 2 21 2 0 z 2 + − () = εε θ sin Qik ik ei zz () = − () + −− αεεθβ εε exp sin exp sin 21 2 21 2 θθ i z () . QTik ee i zz () = − () exp sin εε θ 31 2 εθ εθ εθ 12233 sin sin sin . i == z = 0, .d 11 122 +=+ − () = − () FF eei αβ ε θ εαβ θ ,cos cos αβ εαβ θε ϕϕ ϕ ϕϕ eeTe ee ii e i ii 22 3 22 22 += − () = −− ,cos 333 3 Te e i cos . θ ϕ ϕεθϕεθ 222333 ==kd kdcos , cos . TF1710_book.fm Page 61 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 62 Radio Propagation and Remote Sensing of the Environment Solutions to these equations have the following forms: •For the reflection coefficient from the layer: (3.36a) •For the amplitudes of the directed and reflected waves inside the layer: (3.36b) •For the coefficient of transmission: (3.36c) Here, (3.37) is the reflection coefficient of the horizontally polarized waves from the interface of media 1 and 2, and (3.38) is the corresponding coefficient of the reflection from the interface of media 2 and 3. The same results are obtained for H-waves. It is necessary, in the previous formulae, to substitute the reflection coefficients of horizontally polarized waves for the equivalent ones of vertically polarized waves. We will not analyze the general formulae, as doing so can be rather complicated. Instead, we will confine ourselves to the particular case of a vertical wave incident on the layer when the reflection coefficients for the E- and H-waves are similar. We will suppose for simplicity that ε 1 = 1; that is, assume, for example, that we have a wave incident on the layered ground from the air. Then, (3.39) F FFe FF e ei i eie = () + () + () ( ) 12 23 2 2 12 23 2 2 1 θθ θθ ϕ e ee i2 2 ϕ , α θ βα θ ϕ ϕ = + + () = () 1 1 23 2 23 2 2 2 2 F Fe Fe e e i e i 2 ,, T FF e F e eee i e = + () + ()     + () − () 11 1 23 2 23 2 23 θ θ ϕϕ ee i2 2 ϕ . F i ii i e 12 121 2 121 θ εθεε θ εθεε () = −− + − cos sin cos si nn 2 θ i F e 23 2 2232 2 2 2232 θ εθεε θ εθεε () = −− + − cos sin cos si nn 2 2 θ FF 12 2 2 23 23 23 1 1 = − + = − + ε ε εε εε ,. TF1710_book.fm Page 62 Thursday, September 30, 2004 1:43 PM [...]... 12   23 e −2 τ  1− F  1− F    ( 2 1 + F 12 F 23 e −2 τ − 2 F 12 F 23 e − τ cos 2ψ + ψ 23 ) (3. 42) Although we assume the value ε′′ is small, the absorption coefficient in the layer 2 may be large: τ≅ © 2005 by CRC Press kd ε′′ 2 ε′2 (3. 43) TF1710_book.fm Page 64 Thursday, September 30 , 2004 1: 43 PM 64 Radio Propagation and Remote Sensing of the Environment The frequency dependence of the reflection... September 30 , 2004 1: 43 PM 70 Radio Propagation and Remote Sensing of the Environment described by Equation (3. 31) if somewhere ε(z) < ε0 and angle θi is such that in the same point z0: ( ) ε 0 sin θ i = ε z 0 (3. 68) The point z0 is the turning point, and the total internal wave reflection occurs on the plane z = z0 It is necessary to look for a solution more precise than the WKB approximation for the area... senseless in the given case, but it is more © 2005 by CRC Press TF1710_book.fm Page 66 Thursday, September 30 , 2004 1: 43 PM 66 Radio Propagation and Remote Sensing of the Environment logical compared to their energy flows If the following is the power flow density of the incident wave: c Ei 8π Si = 2 then the flow density of its energy is: c Πi = 8π ∞ ∫ E (t ) i −∞ 2 c dt = 4 ∞ ∫ E (ω ) 2 dω (3. 52) −∞ The Parseval... calculations, the z-axis is directed downward) The reflected signal form is described by a function of the form: ( ) E z,t = ∞ ∫ E (ω ) F (ω ) e − iωt − ikz dω (3. 50) −∞ If we substitute Equation (3. 49) here and assume that the role of the permittivity frequency dispersion is weak in the frame of the signal bandwidth and that the absorption in the layer does not depend on the frequency, then the result of calculations... satisfies the equation: ( ) dW = ik ε − W 2 , dz (3. 75) which is simpler than Equation (3. 71) from many aspects The initial condition for Equation (3. 75) is: ( ) W d = ε∞ , © 2005 by CRC Press (3. 76) TF1710_book.fm Page 72 Thursday, September 30 , 2004 1: 43 PM 72 Radio Propagation and Remote Sensing of the Environment and W= 1 Q′ ik Q (3. 77) replaces nonlinear Equation (3. 75) with a linear one of the second... frequency The amplitude of these oscillations and their quasi-period depend on the layer thickness and its complex permittivity In particular, the amplitude of oscillations decreases with increased absorption and tends to zero for the absolute absorptive layer Let us simplify the problem by considering the case of the dielectric layer The imaginary part of ε2 is small, and ψ12 = π in this case Then, 2... p(z) One of these methods is the Wentzel–Kramers–Brillouin (WKB) method.18,19 It can be employed for the case when the scale of function p(z) changes little compared to wavelength λ The analytical properties of the indicated function are determined © 2005 by CRC Press TF1710_book.fm Page 68 Thursday, September 30 , 2004 1: 43 PM 68 Radio Propagation and Remote Sensing of the Environment by the analytical... permittivity break occurs at the point z = 0 (i.e., ε(z) changes at interface), then the value of function V(0) in Equation (3. 83) should be taken at z = –0 (i.e., left of the break), and the permittivity value in Equation (3. 83) is also taken left of the break Finally, we will consider the case of slow permittivity variation in the layer The WKB method may be used for finding the functions Q1 and Q2, written as:... represent the model parameters by the letter m; that is, we will introduce permittivity εm(z) and the corresponding admittance Wm(z,k) By Equation (3. 78), the function Qm(z,k) satisfies the equation: d 2Q m + k 2ε mQ m = 0 dz 2 © 2005 by CRC Press (3. 122) TF1710_book.fm Page 82 Thursday, September 30 , 2004 1: 43 PM 82 Radio Propagation and Remote Sensing of the Environment We can then multiply Equation (3. 78)... )  2   ℑ = − ln 2 e  − F 23 d e  4 ε1 © 2005 by CRC Press ( ) (3. 1 03) TF1710_book.fm Page 78 Thursday, September 30 , 2004 1: 43 PM 78 Radio Propagation and Remote Sensing of the Environment The requirement for the value obtained to be small is easily reached when a small permittivity jump occurs in the considered point, or observance of the inequality |ε2 – ε1| . inside the layer: (3. 36b) •For the coefficient of transmission: (3. 36c) Here, (3. 37) is the reflection coefficient of the horizontally polarized waves from the interface of media 1 and 2, and (3. 38) is. September 30 , 2004 1: 43 PM © 2005 by CRC Press 60 Radio Propagation and Remote Sensing of the Environment Then, the problem is reduced to solution of the common differential equation: (3. 31) The. 58 Radio Propagation and Remote Sensing of the Environment (3. 23a) (3. 23b) (3. 23c) (3. 23d) Here, Θ r = arg F h – arg F v . Similar relations can be obtained for the refracted